Question

solve the equation. ( e^{x + 2} = 23 ) a 1.135 b 3.045 c 3.219 d 7.25

Explanation:

Step1: Take natural log of both sides

To solve the equation \( e^{x + 2}=23 \), we take the natural logarithm (ln) of both sides. Recall that \( \ln(e^a)=a \) by the property of logarithms and exponential functions (they are inverses). So, applying \( \ln \) to both sides:
\( \ln(e^{x + 2})=\ln(23) \)
Simplifying the left - hand side using the inverse property, we get:
\( x + 2=\ln(23) \)

Step2: Solve for x

Now, we need to isolate \( x \). We know that \( \ln(23)\approx3.135 \) (using a calculator to find the natural logarithm of 23). Then, subtract 2 from both sides of the equation \( x + 2=\ln(23) \):
\( x=\ln(23)-2 \)
Substitute \( \ln(23)\approx3.135 \) into the equation:
\( x\approx3.135 - 2=1.135 \)

Answer:

A. 1.135